Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

Definition 1.1

(Variation operators for singular integrals and their commutators). Let 𝓣_K = {T_K,ϵ}_ϵ>0 and TK,bm={TK,b,ϵm}ϵ>0 with m ≥ 1. For ρ > 2, the ρ-variation operator of 𝓣_K is defined by

Analogously, the ρ-variation operator of TK,bm can be given as

where the above sup is taken over all sequences {ε_i} decreasing to zero. Clearly, Vρ(TK,bm) = 𝓥_ρ (𝓣_K) for m = 0. For convenience, we set Vρ(TK,bm) = 𝓥_ρ (𝓣_K,b) for m = 1.

The operators defined in (1.1) and (1.2) have some classical models, which are listed as follows:

• When n = 1 and K(x,y)=1x−y, then T_K (resp., TK,bm ) is the (resp., the m-th order commutator of) Hilbert transform. We denote 𝓣_K = 𝓗 and TK,bm=Hbm for m ≥ 1.

• When n = 1 and K(x, y) = ℜ^±(x, y), where ℜ^±(x, y) is a Hermit Riesz kernel whose expressions can be found in [32], then T_K (resp., TK,bm ) is (resp., the m-th order commutator of) Hermit Riesz transform. We denote 𝓣_K = 𝓡^± and TK,bm=R±,bm for m ≥ 1.

• When n ≥ 2 and K(x, y) = R_j(x, y), where Rj(x,y):=Γ(n+12)π−n+12xj−yj|x−y|n+1 for 1 ≤ j ≤ n, then T_K (resp., TK,bm ) is (resp., the m-th order commutator of) Riesz transform. We denote 𝓣_K = 𝓡_j and TK,bm=Rj,bm for m ≥ 1.

• When n ≥ 2 and K(x, y) = Ω(x−y)|x−y|n, where Ω ∈ L¹(Sⁿ⁻¹) is homogeneous of zero and satisfies

  ∫Sn−1Ω(θ)dσ(θ)=0. (1.5)

  Then T_K (resp., TK,bm ) is just the usual (resp., the m-th order commutator of) singular integral operator with rough kernel Ω. We denote 𝓣_K = 𝓣_Ω and TK,bm=TΩ,bm for m ≥ 1.

• When T_K is bounded on L²(ℝⁿ) and the kernel K is a standard Calderón-Zygmund kernel, which satisfies the size condition

  |K(x,y)|≤A|x−y|n,forx≠y; (1.6)

  and the regularity conditions for some δ > 0

  |K(x,y)−K(z,y)|≤A|x−z|δ|x−y|n+δ,for|x−y|>2|x−z|; (1.7)

  |K(y,x)−K(y,z)|≤A|x−z|δ|x−y|n+δ,for|x−y|>2|x−z|. (1.8)

Then T_K (resp., TK,bm ) is the (resp., the m-th order commutator of) standard Calderón-Zygmund singular integral operator on ℝⁿ.

Throughout this paper, we always assume that ρ > 2 since the ρ-variation in the case ρ ≤ 2 is often not bounded (see [1, 2]). Let us attribute the developments of the variation operators for singular integrals to two stages.

Stage 1 (n = 1). The variation operators for singular integrals were first studied by Campbell et al. [3] who showed that 𝓥_ρ(𝓗) is of type (p, p) for 1 < p < ∞ and of weak type (1, 1). The above result was later extended to weighted version in [8, 13]. Same conclusions hold for 𝓥_ρ(𝓡^±) (see [8, Theorem A]). A general result was given by Liu and Wu [23] who proved that 𝓥_ρ(𝓣_K) is bounded on L^p(w) for 1 < p < ∞ and w ∈ A_p(ℝ), provided that n = 1 and 𝓥_ρ(𝓣_K) is of type (p₀, p₀) for some p₀ ∈ (1, ∞) and K satisfies the conditions (1.6)-(1.8).

Stage 2 (n ≥ 1). In 2002, Campbell et al. [4] first established the L^p(ℝⁿ) (1 < p < ∞) bounds for 𝓥_ρ(𝓣_Ω), provided that Ω ∈ L log⁺ L(Sⁿ⁻¹). This result was essentially improved by Ding et al. [9] to the case Ω ∈ H¹(Sⁿ⁻¹) since L log⁺ L(Sⁿ⁻¹) ⊊ H¹(Sⁿ⁻¹), which is a proper inclusion. The weighted result for 𝓥_ρ(𝓣_Ω) was first considered by Ma et al. [25] who proved that 𝓥_ρ (𝓣_Ω) is bounded on L^p(w) for 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that Ω ∈ Lip_α(Sⁿ⁻¹) for α > 0. Later on, the above result was improved by Chen et al. [5] to the case Ω ∈ L^q(Sⁿ⁻¹) for some q > 1. In [13], Gillespie and Torrea studied the variation operators for Riesz transforms and showed that 𝓥_ρ(𝓡_j) is bounded on L^p(∣x∣^α) for 1 < p < ∞ and −1 < α < p − 1. Recently, Zhang and Wu [35] extended the above result to general A_p weight. Particularly, Ma et al. [25] proved that 𝓥_ρ(𝓣_K) is bounded on L^p(w) for all 1 < p < ∞ and w ∈ A_p(ℝⁿ) if K satisfies (1.6)-(1.8) and the following priori estimate:

for some p₀ ∈ (1, ∞).

The variation operator for the commutators was first studied by Liu and Wu [23] who showed that Vρ(TK,bm) is bounded on L^p(w) for 1 < p < ∞ and w ∈ A_p(ℝ), provided that m ≥ 1, b ∈ BMO(ℝ) and K satisfies (1.6)-(1.9). As applications, they obtained the L^p(w) bounds for Vρ(Hbm) and Vρ(R±,bm) for 1 < p < ∞ and w ∈ A_p(ℝ) if b ∈ BMO(ℝ). Recently, Liu and Cui [22] extended the above results to the general case n ≥ 1. For the commutator of rough singular integral, Chen et al. [6] proved that Vρ(TΩ,bm) is bounded on L^p(w) for 1 < p < ∞ if Ω ∈ L^q(Sⁿ⁻¹) for some q > 1 satisfying (1.5), m = 1, b ∈ BMO(ℝ) and one of the following conditions holds: (a) q′ ≤ p < ∞, p ≠ 1 and w ∈ A_p/q′(ℝⁿ); (b) 1 < p ≤ q, p ≠ ∞ and w−1p−1 ∈ A_p′/q′(ℝⁿ). Actually, applying the above result, the method in the proof of [6, Theorem 1.1] and induction arguments as in getting [11, Theorem 1], one can conclude that Vρ(TΩ,bm) is bounded on L^p(w) for 1 < p < ∞ if Ω ∈ L^q(Sⁿ⁻¹) for some q > 1 satisfying (1.5), m ≥ 1, b ∈ BMO(ℝ) and one of the following conditions holds: (a) q′ ≤ p < ∞, p ≠ 1 and w ∈ A_p/q′(ℝⁿ); (b) 1 < p ≤ q, p ≠ ∞ and w−1p−1 ∈ A_p′/q′(ℝⁿ). These conclusions together with the main result of [5] imply the following result.

Theorem A

Let m ∈ 𝓝, ρ > 2, b ∈ BMO(ℝⁿ) and Ω ∈ L^q(Sⁿ⁻¹) for some q > 1 satisfying (1.5). Then, for 1 < p < ∞, we have

Recently, Guo et al. [14] first studied the compactness for 𝓥_ρ(𝓣_K,b) on L^p(w). They proved that 𝓥_ρ(𝓣_K,b) is a compact operator on L^p(w) for 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that b ∈ CMO(ℝⁿ) and K satisfies (1.6)-(1.9). Here CMO(ℝⁿ) is the closure of Cc∞ (ℝⁿ) in the BMO(ℝⁿ) topology, which coincides with the space of functions of vanishing mean oscillation. Very recently, Liu and Cui [22] showed that Vρ(TK,bm) is a compact operator on L^p(w) for 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that m ≥ 1, b ∈ CMO(ℝⁿ) and K satisfies (1.6)-(1.9).

Definition 1.2

(Weighed Morrey spaces) ([19]). Let 1 ≤ p < ∞ and 0 ≤ β < 1. For a weight w defined on ℝⁿ, the weighted Morrey space M^p,β(w) is defined by

where

where the supremum is taken over all balls in ℝⁿ. Particularly, the M^p,β(w) is just the classical weighted Lebesgue space L^p(w) when β = 0.

When w ≡ 1, M^p,β(w) reduces to the classical Morrey space M^p,β(ℝⁿ), which was first introduced by Morrey [28] to study the local behavior of solutions to second order elliptic partial differential equations. The weighted Morrey spaces M^p,β(w) were originally introduced by Komori and Shirai [19] who established the bounds for the Hardy-Littlewood maximal operator, fractional integral operator and the Calderón-Zygmund singular integral operator on M^p,β(w). Later on, more and more scholars have devoted to investigating the boundedness of various operators on M^p,β(ℝⁿ) (cf. e.g. [12], [29], [30]).

The boundedness of variation operators of singular integrals and their commutators on Morrey spaces was first studied by Zhang and Wu [36] who proved that 𝓥_ρ(𝓣_K) is bounded on M^p, β}(w) for 0 < β < 1, 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that n = 1 and K satisfies (1.6)-(1.9). Very recently, Liu and Cui [22] studied the boundedness and compactness for variation operators of Calderón-Zygmund singular integrals and their commutators on weighted Morrey spaces. To be more precise, they showed that Vρ(TK,bm) is bounded on M^p,β(w) for 0 < β < 1, 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that m ∈ 𝓝, b ∈ BMO(ℝⁿ) and K satisfies (1.6)-(1.9). They also proved that Vρ(TK,bm) is a compact operator on M^p,β(w) for 0 < β < 1, 1 < p < ∞ and w ∈ A_p(ℝⁿ), provided that m ≥ 1, b ∈ CMO(ℝⁿ) and K satisfies (1.6)-(1.9).

Particularly, Liu and Cui [22] obtained the following result.

Theorem B

([22]). Let Ω ∈ Lip_α(Sⁿ⁻¹) for some α > 0 and Ω satisfy (1.5). Let ρ > 2, 1 < p < ∞ and 0 ≤ β < 1. Then

1. If m ∈ 𝓝 and b ∈ BMO(ℝⁿ), then

   ∥Vρ(TΩ,bm)(f)∥Mp,β(Rn)≤C∥b∥BMO(Rn)m∥f∥Mp,β(Rn),∀f∈Mp,β(Rn).

2. If m ≥ 1 and b ∈ CMO(ℝⁿ), then Vρ(TΩ,bm) is a compact operator on M^p,β(ℝⁿ).

It is well known that

Note that the above inclusion relationship is proper.

Based on Theorem B and (1.10), a natural question is the following

Question 1.3

Does Theorem B hold under the condition that Ω ∈ L^q(Sⁿ⁻¹) for some q > 1?

This is one of the main motivations of this work. In this paper, we shall establish the following results.

Theorem 1.1

Let m ∈ 𝓝, ρ > 2, 0 ≤ β < 1 and 1 < p < ∞. Assume that b ∈ BMO(ℝⁿ) and Ω ∈ L^q(Sⁿ⁻¹) for some q > 1 satisfying (1.5). Then

Theorem 1.2

Let m ≥ 1, ρ > 2, 1 < p < ∞ and 0 ≤ β < 1. Let Ω ∈ L^q (Sⁿ⁻¹) for some q > 1 and satisfy (1.5). For r ≥ 1, define

Here w_r(δ) denotes the integral modulus of continuity of order r of Ω defined by

and ρ is a rotation in ℝⁿ and ∥ρ∥: = sup_{x′∈Sⁿ⁻¹} ∣ρ x′ − x′∣. Assume that b ∈ CMO(ℝⁿ) and F(1) < ∞, then the operator Vρ(TΩ,bm) is a compact operator on M^p,β(ℝⁿ).

Remark 1.1

1. The condition (1.11) was firstly introduced by Chen et al. [7] who proved T_Ω,b is a compact operator on M^p,β(ℝⁿ) for 1 < p < ∞ and 0 < β < 1, provided that Ω satisfies (1.5) and Ω ∈ L^q(Sⁿ⁻¹) for some q > 1/(1 − β) satisfying F(q) < ∞. Note that F(r₁) ≤ CF(r₂) for some C > 0 if r₂ ≥ r₁ ≥ 1.

2. The condition (1.11) is strictly weaker than the condition Ω ∈ Lip_α(Sⁿ⁻¹) with some α > 0. Thus, Theorems 1.1 and 1.2 essentially improve the conclusions of Theorem B.

3. When β = 0, Theorems 1.1 and 1.2 imply the boundedness and compactness of Vρ(TΩ,bm) on the Lebesgue spaces L^p(ℝⁿ).

4. Theorem 1.1 for the case 0 < β < 1 is new, even in the special case m = 0.

5. Theorem 1.2 is new, even in the special case β = 0 and m = 1.

Based on the above, some driving questions are the following

Question 1.5

Do the conclusions in Theorems 1.1 and 1.2 hold under the condition that Ω ∈ L log⁺ L(Sⁿ⁻¹) or Ω ∈ H¹(Sⁿ⁻¹) or Ω ∈ 𝓕_α(Sⁿ⁻¹) for some α > 0?

Proposition 1.3

([23]). Let T be a sublinear operator. Assume that T : L^p(ℝⁿ) → L^p(ℝⁿ) for some p ∈ (1, ∞). If T satisfies

for any x, ζ ∈ ℝⁿ. Here Δ_ζ(f) is the difference of f for an arbitrary function f defined on ℝⁿ and ζ ∈ ℜ_n, i.e., Δ_ζ(f)(x) = f_ζ(x)−f(x) and f_ζ(x) = f(x + ζ). Then T is bounded on B˙sp,q (ℝⁿ) for 0 < s < 1 and 1 < q < ∞. Specially, if T also satisfies the following

for arbitrary functions f, g defined on ℝⁿ. Then T is continuous from Bsp,q (ℝⁿ) to B˙sp,q (ℝⁿ) for 0 < s < 1 and 1 < q < ∞.

Note that the operator 𝓥_ρ(𝓣_K) is sublinearity and commutes with translations, i.e. 𝓥_ρ(𝓣_K)(f)(x + h) = 𝓥_ρ(𝓣_K)(f_h)(x) when K(x, y) = K(x − y). One can easily check that 𝓥_ρ(𝓣_K) satisfies (1.13) and (1.14). Applying Proposition 1.3, we have the following result.

Proposition 1.4

Let ρ > 2 and 𝓥_ρ(𝓣_K) be given as in (1.3). Assume that K(x, y) = K(x − y) and 𝓥_ρ(𝓣_K) is bounded on L^p(ℝⁿ) for some p ∈ (1, ∞). Then 𝓥_ρ(𝓣_K) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1 and 1 < q < ∞.

As applications of Proposition 1.4, the following results are valid.

Corollary 1.5

Let ρ > 2 and 𝓥_ρ(𝓣_K) be given as in (1.3). Assume that K(x, y) = K(x − y) and K satisfies the conditions (1.6)-(1.9). Then 𝓥_ρ(𝓣_K) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1, 1 < p < ∞ and 1 < q < ∞.

Corollary 1.6

Let ρ > 2 and one of the following conditions hold:

1. n = 1 and 𝓣 = 𝓗;

2. n = 1 and 𝓣 = 𝓡^±;

3. n ≥ 2, 𝓣 = 𝓡_j, 1 ≤ j ≤ n;

4. n ≥ 2 and 𝓣 = 𝓣_Ω, where Ω ∈ H¹(Sⁿ⁻¹) or Ω ∈ ⋂_{α > 2} 𝓕_α(Sⁿ⁻¹). Here 𝓕_α(Sⁿ⁻¹) for α > 0 denotes the set of all integrable functions over Sⁿ⁻¹ which satisfy

   supξ′∈Sn−1∫Sn−1|Ω(y′)|(log+⁡1|ξ′⋅y′|)αdσ(y′)<∞.

   Then the operator 𝓥_ρ(𝓣) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1, 1 < p < ∞ and 1 < q < ∞.

Remark 1.6

1. It should be pointed out that Corollary 1.5 follows from Theorem 1 in [25] and Proposition 1.4.

2. The corresponding results in Corollary 1.6 for the cases (i)-(iii) follow from the known bounds for the corresponding operators and Proposition 1.4. It was shown in [9] (see Theorem 1.2 and Corollary 1.6 in [9]) that 𝓥_ρ(𝓣_Ω) is bounded on L^p(ℝⁿ) for all p ∈ (1, ∞) under the condition that Ω ∈ H¹(Sⁿ⁻¹) or Ω ∈ ⋂_{α > 2} 𝓕_α(Sⁿ⁻¹). This together with Proposition 1.4 yields the conclusion of Corollary 1.6 for case (iv).

3. We remark that the space 𝓕_α(Sⁿ⁻¹) was introduced by Grafakos and Stefanov [15] in the study of L^p boundedness of singular integral operator with rough kernels. Clearly,$\bigcup_{q > 1} L^q(Sⁿ⁻¹) ⊊ 𝓕_α(Sⁿ⁻¹) for any α > 0. Moreover, the examples in [15] show that

   ⋂α>1Fα(Sn−1)⊈H1(Sn−1)⊈⋃α>1Fα(Sn−1).

It should be pointed out that the operator Vρ(TK,bm) does not satisfy the condition (1.13), even in the special case m = 1 and K(x, y) = K(x − y), which makes that Proposition 1.3 does not apply for Vρ(TK,bm) Therefore, it is natural to ask the following

Question 1.7

Is the operator Vρ(TK,bm) bounded and continuous on Bsp,q (ℝⁿ) for some 0 < s < 1 and 1 < p, q < ∞ when m ≥ 1?

In this paper we shall present a positive answer to this question, which is another one of main motivations. Before presenting the rest of main results, let us introduce the following definition.

Definition 1.8

(Lipschitz space). The homogeneous Lipschitz space Λ̇(ℝⁿ) is given by

where

The inhomogeneous Lipschitz space Λ(ℝⁿ) is defined by

The rest of main results can be formulated as follows:

Proposition 1.7

Let ρ > 2, m ≥ 1 and Vρ(TK,bm) be given as in (1.4). Assume that b ∈ Λ(ℝⁿ), K(x, y) = K(x − y) and 𝓥_ρ(𝓣_K) is bounded on L^p(ℝⁿ) for some p ∈ (1, ∞). Then Vρ(TK,bm) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1 and 1 < q < ∞. Particularly,

As some applications of Proposition 1.7, we obtain

Corollary 1.8

Let ρ > 2, m ≥ 1 and Vρ(TK,bm) be given as in (1.4). Assume that b ∈ Λ(ℝⁿ), K(x, y) = K(x − y) and K satisfies the conditions (1.6)-(1.9). Then Vρ(TK,bm) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1, 1 < p < ∞ and 1 < q < ∞. Particularly,

Corollary 1.9

Let m ≥ 1, ρ > 2, b ∈ Λ(ℝⁿ) and one of the following conditions hold:

1. n = 1 and T=Hbm;

2. n = 1 and T=R±,bm;

3. T=Rj,bm, 1 ≤ j ≤ n;

4. T=TΩ,bm, where Ω ∈ H¹(Sⁿ⁻¹) or Ω ∈ ⋂_{α > 2} 𝓕_α(Sⁿ⁻¹).

Then the operator 𝓥_ρ(𝓣) is bounded and continuous on Bsp,q (ℝⁿ) for 0 < s < 1, 1 < p < ∞ and 1 < q < ∞. Moreover,

Remark 1.9

1. Corollary 1.8 follows from Theorem A and Proposition 1.7.

2. The corresponding results in Corollary 1.9 for the cases (i)-(iii) follow from the known bounds for the corresponding operators and Proposition 1.7. It was shown in [9] (see Theorem 1.2 and Corollary 1.6 in [9]) that 𝓥_ρ(𝓣_Ω) is bounded on L^p(ℝⁿ) for all p ∈ (1, ∞) under the condition that Ω ∈ H¹(Sⁿ⁻¹) or Ω ∈ ⋂_{α > 2} 𝓕_α(Sⁿ⁻¹). This together with Proposition 1.7 yields the conclusions of Corollary 1.9 for case (iv).

Some interesting questions can be formulated as follows:

Question 1.10

Do the corresponding results in Propositions 1.4 and 1.7 and Corollaries 1.5 and 1.8 hold when K(x, y) ≠ K(x − y)?